An equality of Monge-Amp\`ere measures

TL;DR

This paper proves that if two plurisubharmonic functions agree on a plurifinely open set, then their Monge-Ampère measures are equal on that set, extending previous results to more general measurable sets.

Contribution

It generalizes the equality of Monge-Ampère measures for plurisubharmonic functions to the case where they agree on a Borel measurable plurifinely open set.

Findings

Monge-Ampère measures coincide on plurifinely open sets where functions agree

Extends previous bounded and finite case results to measurable sets

Unifies various special cases under a general theorem

Abstract

Let $u$ and $v$ be two plurisubharmonic functions in the domain of definition of the Monge-Amp\`ere operator on a domain $\Omega\subset {\bf C}^n$. We prove that if $u=v$ on a plurifinely open set $U\subset \Omega$ that is Borel measurable, then $(dd^cu)^n|_U=(dd^cv)^n|_U$. This result was proved by Bedford and Taylor in the case where $u$ and $v$ are locally bounded, and by El Kadiri and Wiegerinck when $u$ and $v$ are finite, and by Hai and Hiep when $U$ is of the form $U=\bigcup_{j=1}^m\{\varphi_j>\psi_j\}$, where $\varphi_j$, $\psi_j$, $j=1,...,m$, are plurisubharmonic functions on $\Omega$.